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The perturbative path-integral gives a morphism of the (quantum) $A_{\infty}$ structure intrinsic to each quantum field theory, which we show explicitlyon the basis of the homological perturbation. As is known, in the BV formalism,any effective action also solves the BV master equation, which implies that thepath-integral can be understood as a morphism of the BV differential. Sinceeach solution of the BV master equation is in one-to-one correspondence with a(quantum) $A_{\infty }$ structure, the path-integral preserves this intrinsic$A_{\infty }$ structure of quantum field theory, where $A_{\infty }$ reduces to$L_{\infty }$ whenever multiplications of space-time fields are gradedcommutative. We apply these ideas to string field theory and (re-)derive somequantities based on the perturbative path-integral, such as effective theorieswith finite $\alpha ^{\prime }$, reduction of gauge and unphysical degrees,$S$-matrix and gauge invariant observables.

Perturbative path-integral of string field and the $A_{\infty }$ structure of the BV master equation